Problem: A certain company's main source of income is selling cloth bracelets. The company's annual profit (in thousands of dollars) as a function of the price of a bracelet (in dollars) is modeled by: $P(x)=-2x^2+16x-24$ What is the maximum profit that the company can earn?
The company's profit is modeled by a quadratic function, whose graph is a parabola. The maximum profit is reached at the vertex. So in order to find the maximum profit, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $P(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} P(x)&=0 \\\\ -2x^2+16x-24&=0 \\\\ x^2-8x+12&=0 \\\\ (x-2)(x-6)&=0 \\\\ \swarrow &\searrow \\\\ x-2=0\text{ or }&x-6=0 \\\\ x={2}\text{ or }&x={6} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({2})+({6})}{2}=\dfrac{8}{2}= 4$ The vertex's $x$ -coordinate is $ 4$. Now let's find $P({4})$ : $\begin{aligned} P( 4)&=-2( 4)^2+16( 4)-24 \\\\ &=-32+64-24 \\\\ &=8 \end{aligned}$ In conclusion, the maximum profit is $8$ thousand dollars.